Scilab Function
Last update : April 1993

sfact - discrete time spectral factorization

### Calling Sequence

  `F=sfact(P)`

### Parameters

- `P`: real polynomial matrix

### Description

Finds `F`, a spectral factor of `P`. `P` is a polynomial matrix such that each root of `P` has a mirror image w.r.t the unit circle. Problem is singular if a root is on the unit circle.

`sfact(P)` returns a polynomial matrix `F(z)` which is antistable and such that

`P = F(z)* F(1/z) *z^n`

For scalar polynomials a specific algorithm is implemented. Algorithms are adapted from Kucera's book.

### Examples

    //Simple polynomial example
    z=poly(0,'z');
    p=(z-1/2)*(2-z)
    w=sfact(p);
    w*numer(horner(w,1/z))
    //matrix example
    F1=[z-1/2,z+1/2,z^2+2;1,z,-z;z^3+2*z,z,1/2-z];
    P=F1*gtild(F1,'d');  //P is symmetric
    F=sfact(P)
    roots(det(P))
    roots(det(gtild(F,'d')))  //The stable roots
    roots(det(F))             //The antistable roots
    clean(P-F*gtild(F,'d'))
    //Example of continuous time use
    s=poly(0,'s');
    p=-3*(s+(1+%i))*(s+(1-%i))*(s+0.5)*(s-0.5)*(s-(1+%i))*(s-(1-%i));p=real(p);
    //p(s) = polynomial in s^2 , looks for stable f such that p=f(s)*f(-s)
    w=horner(p,(1-s)/(1+s));  // bilinear transform w=p((1-s)/(1+s))
    wn=numer(w);              //take the numerator
    fn=sfact(wn);f=numer(horner(fn,(1-s)/(s+1))); //Factor and back transform
    f=f/sqrt(horner(f*gtild(f,'c'),0));f=f*sqrt(horner(p,0));      //normalization
    roots(f)    //f is stable
    clean(f*gtild(f,'c')-p)    //f(s)*f(-s) is p(s)

### See Also

`gtild`, `fspecg`,